1- Many high school students take the AP tests in different subject areas. In 2017, of the 144,790 students who took the biology exam 84,200 of them were female. In that same year, of the 211,693 students who took the calculus AB exam 102,598 of them were female. Is there enough evidence to show that the proportion of female students taking the biology exam is higher than the proportion of female students taking the calculus AB exam? Test at the 5% level.
Hypotheses
\(H_0\): p1 <= p2 \(H_a\): p1 > p2
where:
\(p_1\)= This is the proportion of the female students that took the Biology Exam.
\(p_2\)= This is the proportion of female students that took Calculus AB Exam.
Significance Level
α = 0.05
p-value
prop.test(x = c(84200,102598), n = c(144790,211693), alternative = "greater", correct = FALSE)
2-sample test for equality of proportions without continuity correction
data: c(84200, 102598) out of c(144790, 211693)
X-squared = 3235.3, df = 1, p-value < 2.2e-16
alternative hypothesis: greater
95 percent confidence interval:
0.09409523 1.00000000
sample estimates:
prop 1 prop 2
0.5815319 0.4846547 Science the p-value 0.05, we reject the null hypotheses.
There is a strong statistical evidence that the proportion of the female students taking the AP Biology exam is higher than the proportion of the female students taking the AP Calculus AB exam.
2- A vitamin K shot is given to infants soon after birth. The study is to see if how they handle the infants could reduce the pain the infants feel. One of the measurements taken was how long, in seconds, the infant cried after being given the shot. A random sample was taken from the group that was given the shot using conventional methods, and a random sample was taken from the group that was given the shot where the mother held the infant prior to and during the shot. Is there enough evidence to show that infants cried less on average when they are held by their mothers than if held using conventional methods? Test at the 5% level.
State the Hypotheses Test
\(H_0\): \(\mu_1\) \(H_a\): \(\mu_2\)
where:
\(\mu_1\) = mean of crying time using conventional holding
\(\mu_2\) = Holding infant with the mother
State the Significance Level
α = 0.05
p-value
conventional_methods <- c(63,0,2,46,33,33,29,23,11,12,48,15,33,14,51,37,24,70,63,0,73,39,54,52,39,34,30,55,58,18)
new_methods <- c(0,32,20,23,14,19,60,59,64,64,72,50,44,14,10,58,19,41,17,5,36,73,19,46,9,43,73,27,25,18)t.test(new_methods, conventional_methods, alternative = "less", paired = FALSE)
Welch Two Sample t-test
data: new_methods and conventional_methods
t = -0.029953, df = 57.707, p-value = 0.4881
alternative hypothesis: true difference in means is less than 0
95 percent confidence interval:
-Inf 9.135003
sample estimates:
mean of x mean of y
35.13333 35.30000 Because the p-value (0.488) is greater than significance level of 0.05, we fail to reject H0.
There is not enough evidence at the 5% significance level to conclude that infants cry less on average when they are held by their mothers compared to when held using conventional methods.